The Extension Theorem Original Research Article

Given a compact convex polyhedron, can it tile space in a transitive (or regular) way? We discuss here the Extension Theorem, which gives conditions under which there is unique extension of a finite polyhedral complex (replicas of the given polyhedron) to a global isohedral tiling. The extension theorem gives a way to get all possible regular tilings with the given polyhedron. The well-known results on fundamental domains in the case of a translation group or a Coxeter group generated by mirrors follow from the extension theorem too. The extension theorem also gives a method of describing which finite point sets can admit extension to a regular point orbits with respect to crystallographic groups.